Hadamard expansions for powers of causal Green’s operators and “resolvents”

Abstract

The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with $P-z$ for $z\in \mathbb {C}$. These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions.